Stability Theorems for H-Type Carnot Groups

Abstract

The H-type deviation, δ(G), of a step two Carnot group G quantifies the extent to which G deviates from the geometrically and algebraically tractable class of Heisenberg-type (Htype) groups. In an earlier paper, the author defined this notion and used it to provide new analytic characterizations for the class of H-type groups. In addition, a quantitative conjecture relating the H-type deviation to the behavior of the ∞-Laplacian of Folland's fundamental solution for the 2-Laplacian was formulated; an affirmative answer to this conjecture would imply that all step two polarizable groups are of H-type. In this paper, we elucidate further properties of the H-type deviation. First, we show that 0 ≤ δ(G) ≤ 1 for all step two Carnot groups G. Recalling that δ(F2,m) = (m − 2)/m, where F2,m is the free step two Carnot group of rank m, we conjecture that δ(G) ≤ (m − 2)/m for all step two rank m groups. We explicitly compute δ(G) when G is a product of Heisenberg groups and verify the conjectural upper bound for such groups, with equality if and only if G factors over the first Heisenberg group. We also prove the following rigidity statement: for each m ≥ 3 there exists δ0(m) > 0 so that if G is a step two and rank m Carnot group with δ(G) < δ0(m), then G enjoys certain algebraic properties characteristic of H-type groups.